class: title-slide background-image: url("https://raw.githubusercontent.com/garthtarr/sydney_xaringan/master/assets/USydLogo-white.svg"), url("https://raw.githubusercontent.com/garthtarr/sydney_xaringan/master/assets/title-image1.jpg") background-position: 10% 90%, 100% 50% background-size: 160px, 50% 100% background-color: #0148A4 # .text-shadow[.white[STAT5003]] ## .white[Week 3 : Density Estimation] ### .white[Dr. Justin Wishart] ### .white[Semester 2, 2020] --- ## Readings <i class="fas fa-university fa-pull-right "></i> * For the bias variance tradeoff see Section 2.2 James, Witten, Hastie, and Tibshirani (2013) --- class: segue-yellow background-image: url("https://raw.githubusercontent.com/garthtarr/sydney_xaringan/master/assets/USydLogo.svg") background-size: 20% background-position: 95% 95% # Review on probability distribution functions --- ## Discrete distributions For any random variable `\(X\)` with a discrete distribution, there is a sample space `\(\Omega\)` with finite number of possible values (outcomes) `\(x = \left\{ x_1, x_2, \ldots \right\}\)` and associated probabilities `\(\left\{p_1, p_2, \ldots \right\}\)`. The point probabilities for each value of `\(x\)` are denoted `\(f(x)\)` and the cumulative distribution function denoted `\(F(x)\)` where `\begin{align*} f(x) = P(X = x), \qquad F(x) = P(X \le x) \end{align*}` Properties: * There is a _countable_ number of possible values; * `\(\sum_{i = 1}^\infty p_i = 1\)` * `\(p_i \ge 0\)` --- ## Binomial distribution .columns-2[ .pull-left[ `\begin{align*} f(x) = \begin{cases} \binom{n}{x} p^x (1 - p)^{n - x}, & x = 0, 1, 2, \ldots\\ 0,& \text{otherwise}\end{cases} \end{align*}` The `\(\binom{n}{x}\)` are known as the binomial coefficients. The parameter `\(p\)` is the probability of success. ```r x <- 0:50 prob <- dbinom(x, size = 50, prob = 0.33) # Base R graphics plot(x, prob, type = "h") dat <- data.frame(x = x, y = prob) # ggplot2 version ggplot(dat, aes(x = x, y = y, xend = x, yend = 0)) + geom_segment() + theme_minimal() ``` ] .pull-right[ <img src="data:image/svg+xml;base64,<?xml version='1.0' encoding='UTF-8' ?>
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" width="80%" /> ]] --- ## Continuous distributions * A continuous random variable `\(X\)` is where the outcome can take an infinite (uncountable) number of possible values. - These values may be within a fixed or unbounded interval. * For example, the height of male in cm may be within the range of [50, 300]. The point probabilities for each value of `\(x\)` is `\(P(X = x) = 0\)` and the cumulative distribution function `\begin{align*} F(x) = \int_{-\infty}^x f(t) \, dt = P(X \le x) \end{align*}` Properties: * There are an infinite (uncountable) number of possible values; * `\(f(x)\)` is called the density function * `\(f(x) \ge 0\)` (non-negative) * `\(\displaystyle \int_{-\infty}^\infty f(x) \, dx = 1\)` (unit measure) --- ## Normal(Gaussian) distribution: `\(\displaystyle f(x) = \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x - \mu)^2}{2\sigma^2}}\)` .columns-2[ .pull-left[ * The most famous continuous distribution * Fully specified by two parameters - `\(\mu\)` the location parameter (mean) - `\(\sigma\)` the scale parameter (sd) * Notation `\(X \sim \mathcal N(\mu, \sigma)\)`, ```r mu <- 0; sig <- 1 x <- seq(from = mu - 4 * sig, to = mu + 4 * sig, length.out = 128) dens <- dnorm(x, mean = mu, sd = sig) # Base R graphics plot(x, dens, type = "l") dat <- data.frame(x = x, y = dens) # ggplot2 version ggplot(dat, aes(x = x, y = y)) + geom_line() + theme_minimal() ``` ] .pull-right[ <img src="data:image/svg+xml;base64,<?xml version='1.0' encoding='UTF-8' ?>
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" width="80%" /> ]] --- class: segue-yellow background-image: url("https://raw.githubusercontent.com/garthtarr/sydney_xaringan/master/assets/USydLogo.svg") background-size: 20% background-position: 95% 95% # Density estimation --- ## Density estimation In exploratory data analysis, an estimate of the density function can be used * to assess multimodality, skew, tail behaviour, etc. * in decision making, classification, and summarizing Bayesian posteriors * as a useful visualisation tool (a simple summary of a distribution) Suppose random variables `\(X_1, X_2, \ldots, X_n\)` have been observed and assumed to be sampled independently from the distribution with density `\(f\)`. __Goal__: The estimation of the density function `\(f\)`. --- ## Parametric density estimation * The .brand-red[parametric] approach to density estimation assumed a .brand-red[parametric] model. * That is, `\(X_1, X_2, \ldots, X_n \stackrel{i.i.d.}{\sim } f_\boldsymbol{\theta}\)` where `\(\boldsymbol{\theta}\)` is a parameter vector. - For example, `\(\boldsymbol{\theta} = (\mu, \sigma)\)` when `\(X \sim \mathcal N(\mu, \sigma)\)` * Typically the parameter `\(\boldsymbol{\theta}\)` is estimated using the method of .brand-red[maximum likelihood]. * Density function is then estimated as `\(f(x|\widehat{\boldsymbol{\theta}})\)` Maximum likelihood the best value for the parameters is the one for which the probability of obtaining the observed samples is the largest. --- ## What is a likelihood? <div id="htmlwidget-DiagrammeR-1f029ea969ca298b40f5" style="width:70%;height:210px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-DiagrammeR-1f029ea969ca298b40f5">{"x":{"diagram":"digraph G\n{\n rankdir = LR;\n \n node[shape = box, label = \"Population\"]\n A\n\n node[shape = box, label = \"Sample\"]\n B\n \n edge[label = \"Probability\"]\n A -> B \n \n edge[label = \"Likelihood\"]\n B -> A \n}\n","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> Simple example: * Population has girl:boy ratio of 2:1 (100 girls for 50 boys) * If I draw a sample of 50 people, what is the .brand-red[probability] of picking 10 boys * If I draw a sample of 50 people, and picked 10 boys, what is the .brand-red[likelihood] that the girl:boy ratio is 2:1 --- ## Normal distribution example * Consider a random variable `\(X \sim \mathcal N(3.5, 0.2)\)` * What is the probability that `\(X\)` is between `\(3.5\)` and `\(4\)`? - Compute the area under the density. `\(P(3.5 \le X \le 4) = \int_{3.5}^4f(t)\, dt\)` .columns-2[ .pull-left[ ```r mu = 3.2; sig = 0.5 pnorm(4, mean = mu, sd = sig) - pnorm(3.5, mean = mu, sd = sig) ``` ``` ## [1] 0.2194538 ``` ```r # Or in one line ## diff(pnorm(c(3.5, 4), mean = mu, sd = sig)) ``` ] .pull-right[ <img src="data:image/svg+xml;base64,<?xml version='1.0' encoding='UTF-8' ?>
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" width="70%" height="50%" /> * Preferable to have a smooth estimate and not have columns --- ## Kernel functions * A kernel is a special type of .brand-red[probability density function (PDF)] having the properties. - non-negative `\(K(x) \ge 0\)`, symmetric `\(K(-x) = K(x)\)`, unit measure `\(\int K(x) \, dx = 1\)` <img src="data:image/svg+xml;base64,<?xml version='1.0' encoding='UTF-8' ?>
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" width="70%" /> --- ## Kernel density esimation * Kernel density estimation is a non-parametric approach estimating densities - Knowledge of the structure of `\(f\)` is not required * Essentially, at every data point, a kernel function is created with the point at its centre. * The PDF is estimated by adding all of these kernel functions and dividing by the number of data to ensure that it satisfies - every possible value of the PDF is non-negative. - the definite integral of the PDF over its support set equals 1 --- ## Normal kernel density estimate <img src="data:image/svg+xml;base64,<?xml version='1.0' encoding='UTF-8' ?>
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" width="70%" /> * E.g. Four sampled variables marked in .brand-red[red] with Gaussian weights sum together to give the overall density estimate --- ## Kernel density estimator (KDE) * A simple one weights all points within a window `\(h\)` of `\(x\)` equally `\begin{align*} \widehat f(x) = \frac{1}{2nh} \sum_{i = 1}^n \mathbb{1}_{\left\{ |X_i - x| < h \right\}} \end{align*}` * More generally a univariate kernel density estimator has a general weight function (Kernel) `\begin{align*} \widehat f(x) = \frac{1}{nh} \sum_{i = 1}^n K \left( \frac{X_i - x}{h}\right) \end{align*}` - `\(K\)` is a Kernel function - `\(h\)` is a bandwidth parameter (possibly fixed or varying) - Consider only `\(h\)` fixed for this course. --- ## Tuning the Kernel density estimator (KDE) * There are two main components for the KDE `\(\displaystyle \widehat f(x) = \frac{1}{nh} \sum_{i = 1}^n K \left( \frac{X_i - x}{h}\right)\)` - The choice of `\(K\)` - The choice of `\(h\)` * The choice of Kernel is less important and generally gives similar results * The choice of bandwidth is important and can vary the result greatly. * Some standard kernels `\begin{align*} \text{Uniform} & \qquad K(x) = \tfrac{1}{2} \mathbb{1}_{\left\{ |x| \le 1\right\}}\\ \text{Gaussian} &\qquad K(x) = \tfrac{1}{\sqrt{2 \pi}} \exp \left\{ - \tfrac{x^2}{2} \right\}\\ \text{Epanechnikov} & \qquad K(x) = \tfrac{3}{4} (1 - x^2) \mathbb{1}_{\left\{ |x| \le 1\right\}}\\ \end{align*}` --- ## Different choices of Kernel function with same bandwidth <img src="data:image/svg+xml;base64,<?xml version='1.0' encoding='UTF-8' ?>
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" width="70%" height="100%" /> --- ## Computing density in <svg style="height:0.8em;top:.04em;position:relative;fill:steelblue;" viewBox="0 0 581 512"><path d="M581 226.6C581 119.1 450.9 32 290.5 32S0 119.1 0 226.6C0 322.4 103.3 402 239.4 418.1V480h99.1v-61.5c24.3-2.7 47.6-7.4 69.4-13.9L448 480h112l-67.4-113.7c54.5-35.4 88.4-84.9 88.4-139.7zm-466.8 14.5c0-73.5 98.9-133 220.8-133s211.9 40.7 211.9 133c0 50.1-26.5 85-70.3 106.4-2.4-1.6-4.7-2.9-6.4-3.7-10.2-5.2-27.8-10.5-27.8-10.5s86.6-6.4 86.6-92.7-90.6-87.9-90.6-87.9h-199V361c-74.1-21.5-125.2-67.1-125.2-119.9zm225.1 38.3v-55.6c57.8 0 87.8-6.8 87.8 27.3 0 36.5-38.2 28.3-87.8 28.3zm-.9 72.5H365c10.8 0 18.9 11.7 24 19.2-16.1 1.9-33 2.8-50.6 2.9v-22.1z"/></svg> * Base <svg style="height:0.8em;top:.04em;position:relative;fill:steelblue;" viewBox="0 0 581 512"><path d="M581 226.6C581 119.1 450.9 32 290.5 32S0 119.1 0 226.6C0 322.4 103.3 402 239.4 418.1V480h99.1v-61.5c24.3-2.7 47.6-7.4 69.4-13.9L448 480h112l-67.4-113.7c54.5-35.4 88.4-84.9 88.4-139.7zm-466.8 14.5c0-73.5 98.9-133 220.8-133s211.9 40.7 211.9 133c0 50.1-26.5 85-70.3 106.4-2.4-1.6-4.7-2.9-6.4-3.7-10.2-5.2-27.8-10.5-27.8-10.5s86.6-6.4 86.6-92.7-90.6-87.9-90.6-87.9h-199V361c-74.1-21.5-125.2-67.1-125.2-119.9zm225.1 38.3v-55.6c57.8 0 87.8-6.8 87.8 27.3 0 36.5-38.2 28.3-87.8 28.3zm-.9 72.5H365c10.8 0 18.9 11.7 24 19.2-16.1 1.9-33 2.8-50.6 2.9v-22.1z"/></svg> there is `density` - `density` computes the KDE - Can wrap in `plot` (`plot(density(x)))`) to visualize - Can inspect details in `summary` * For plotting ggplot there is `geom_density` - Can specify the bandwidth with `bw` argument --- ## Choosing the bandwidth * The density estimator $$ \widehat f(x) = \frac{1}{nh} K \left( \frac{X_i - x}{h}\right) $$ - is a fixed-bandwidth kernel density estimator since `\(h\)` is constant. * If `\(h\)` is too small, the density estimator will tend to assign probability density too locally near observed data - a wiggly estimated density function with many false modes. * If `\(h\)` is too large, the density estimator will spread probability density contributions too diffusely - smooths away important features of `\(f\)` --- ## Choice of bandwidth * Consider the distance from CBD variable again with three bandwidths .columns-2[ .pull-left[ <img src="data:image/svg+xml;base64,<?xml version='1.0' encoding='UTF-8' ?>
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" width="60%" /> --- ## Mean squared error, Bias and Variance We can decompose the mean squared error (MSE) into the sum of three quantities: The .brand-red[variance], the .brand-red[squared bias] and the .brand-red[vairance of the error]. $$ \mathbb{E} \left(Y - \widehat f(X) \right)^2 = Var(\widehat f(X)) + \left[Bias(\widehat f(X)) \right]^2 + Var(\epsilon) $$ * Variance here denoting how much would `\(\widehat f(x)\)` change if we estimate using a different training set. * Bias - Error introduced by approximating the data using a model. ### Kernel density estimation type equivalent `\begin{align*} Var(\widehat f(x)) &= \mathcal O\left(\frac{1}{nh}\right)\\ Bias(\widehat f(x)) &= \mathcal O(h) \end{align*}` --- ## References James, G, D. Witten, T. Hastie, et al. (2013). _An introduction to statistical learning_. Vol. 112. Springer.